EDEXCEL International Advanced Level
Edexcel IAL Physics Unit 4
Summary©
HASAN SAYGINEL
HS
Further Mechanics
1 MOMENTUM
Momentum is a useful measure of motion, which is defined mathematically as the product
of mass and velocity:
Momentum = mass x velocity
𝑝=𝑚×𝑣
Momentum is a vector quantity and always has both magnitude and direction.
The momentum of an object is a measure of the accelerating force, applied over a period
of time that is needed to bring the object from rest to the speed at which it is moving. An
object’s momentum is also the force required, over a period of time, to bring the moving
object to rest.
2 NEWTON’S SECOND LAW
Newton’s law, F=ma only holds true if the mass remains constant. A more precise way in
which this law is explained is:
The rate of change of momentum of a body is directly proportional to the resultant force
applied to the body, and is in the same direction as the force.
This can be written mathematically as:
𝐹=
The
𝑑(𝑥)
𝑑𝑡
𝑑𝑝 𝑑(𝑚𝑣)
=
𝑑𝑡
𝑑𝑡
term is a mathematical expression meaning the rate of change of x, or how quickly x
changes. If the quantities are not constantly changing, we can express this as:
𝐹=
∆𝑝
∆𝑡
The product of a force applied for a certain time is known as the impulse.
𝑖𝑚𝑝𝑢𝑙𝑠𝑒 = 𝐹 × ∆𝑡 = ∆𝑝
To stop something moving, we need to remove all of its momentum. This idea allows us to
calculate the impulse needed to stop a moving object. And if we know for how long a force is
applied, we can work out the size of that force which ought to be negative as it is in the
reverse direction.
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3 NEWTON’S THIRD LAW
Conservation of momentum is directly responsible for this law. It tells us that for every force,
there is an equal and opposite force. If we think of a force as a means to change momentum,
then a force changing momentum in one direction must be countered by an equal and
opposite force to ensure that overall momentum is conserved.
4 COLLISIONS
Conservation of linear momentum
The total momentum of a system remains constant provided that no external forces act on
the system. This applies to all objects moving in a straight line. This tells us that if we
calculate the momentum of each body before the collision, the sum total of these momenta
accounting for their direction will be the same as the sum total afterwards.
This principle is dependent on the condition that no external force acts on the objects in
question. An external force would provide an additional acceleration, so the motion of the
objects would not be dependent on the collision alone.
Explosions
If a stationary object explodes, then the total momentum of all the shrapnel parts added
up must be zero. The object had zero momentum at the start, so the principle of
conservation of linear momentum tells us this must be the same total after the explosion.
Conservation of linear momentum in 2D
When the objects are moving along different lines, we can resolve the velocity vectors into
perpendicular directions and carry out the momenta sums for the component directions.
5 ENERGY IN COLLISIONS
Elastic collisions
A collision in which kinetic energy is conserved is called an elastic collision. In life, these are
rare. A collision caused by non-contact forces, such as alpha particles being scattered by a
nucleus is perfectly elastic.
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Inelastic collisions
A collision in which kinetic energy is not conserved is called an inelastic collision. Some of the
kinetic energy is converted into other forms such as heat and sound.
6 PARTICLE COLLISIONS
1
We know that the formula for calculating kinetic energy is 𝐸𝑘 = 2 𝑚𝑣 2 and that the formula
for momentum is 𝑝 = 𝑚𝑣. By combining these relationships we can get an equation that
gives kinetic energy in terms of the momentum and mass.
This is a particularly useful formula when dealing with the kinetic energy of subatomic
particles travelling at non-relativistic speeds.
This formula offers an alternative way of calculating the de Broglie wavelength for a particle
if we know its energy and mass:
7 CIRCULAR MOTION
Angular displacement
When we are measuring rotation, we often use an alternative measure of angle called the
radian. The angle, through which the object moves, measured in radians, is defined as the
distance it has travelled along the circumference divided by its distance from the centre of
circle.
𝜃=
𝑠
𝑟
𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 180°
Angular displacement is the vector measurement of the angle through which something has
turned. The standard convention is that anticlockwise rotation is a positive number and
clockwise rotation is a negative number.
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Angular velocity
The rate which the angular displacement changes, is called the angular velocity, w. so,
constant angular velocity is defined mathematically by:
𝜔=
𝜃
𝑡
If the object completes a full circle in a time period T, then the angular is given by:
𝜔=
2𝜋
𝑇
The frequency of rotation is the reciprocal of the time period:
𝑓=
1
𝑇
So:
𝜔 = 2𝜋𝑓
Instantaneous velocity
𝑠
𝑠
We know that 𝑣 = 𝑡 and from the definition of the angle in radians 𝜃 = 𝑟, so that 𝑠 = 𝑟𝜃.
Thus:
𝑣=
𝑟𝜃
𝑡
𝑣 = 𝑟𝜔
8 CENTRIPETAL FORCE
As an object such as a hammer is whirled at a constant speed, the magnitude of the velocity
is always the same. However, the direction the velocity is constantly changing and a change
in velocity is an acceleration. Newton’s first law tells us that acceleration will only happen if
there is a resultant force. The hammer is constantly being pulled towards the centre of the
circle. For any object moving in a circle, there must be a force to cause this acceleration
toward the centre of the circle – this is called the centripetal force. If a force causes
something to move in a circle, we identify it as the centripetal force for that circling object.
The resultant centripetal force needed will be larger if:
The rotating object has more mass
The object rotates faster
The object is closer to the centre of the circle
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Centripetal force and acceleration
The mathematical formula for the centripetal force on an object moving in a circle is:
𝑚𝑣 2
𝐹=
𝑟
Using the relationship, 𝑣 = 𝑟𝜔, we can derive an alternative for centripetal force in terms of
angular velocity:
Since Newton’s second law states that the resultant force is related to the acceleration it
causes by the equation, 𝐹 = 𝑚𝑎, we can find the centripetal acceleration very easily:
Again using, 𝑣 = 𝑟𝜔, the centripetal acceleration can be expressed in terms of the angular
velocity as:
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Electric and magnetic fields
1 ELECTRIC FIELDS
Electric field is a region in space in which a charged object experiences a force.
To visualize the forces caused by the field, we draw electric field lines which show the
direction in which a positively charged particle will be pushed by the force the field
produces. Like all field patterns the closer the lines are together, the stronger the field is.
The force that a charged particle will feel is the electric field strength (E) multiplied by the
amount of charge on the particle in coulombs (Q), as given by the equation:
𝐹 = 𝐸𝑄
From this force equation, we can also quickly calculate how a charge would accelerate.
Newton`s second law states that 𝐹 = 𝑚𝑎, so we can equate the equations for force:
𝐹 = 𝐸𝑄 = 𝑚𝑎
So:
𝑎=
𝐸𝑄
𝑚
Electric field strength E is a vector quantity. The direction of E is the same as the direction of
the electric force F, which is defined as the force on a positive charge.
2 POTENTIAL DIFFERENCE AND ELECTRIC FIELDS
For a charged particle moving in an electric field, the change in its kinetic energy is provided
in a transfer from the electric potential energy the electron had by virtue of its location
within the electric field. Every location within a field has a certain potential. The different
between the potential at an electron`s original location and the potential at a new location
is the potential difference through which the electron moves.
Potential difference is defined as the energy transferred per coulomb of charge passing
through the device. In an electric field, we can follow exactly the same idea in order to find
out how much kinetic energy a charged particle will gain by moving within the field. This is
given by the equation:
𝐸 = 𝑉𝑄
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3 UNIFORM ELECTRIC FIELDS
An electric field exists between any objects which
are at a different electric potential. Two oppositely
charged plates placed parallel to each other produce
a uniform electric field between. In a uniform electric
field, the electric lines of force are equally spaced.
When you move a small charged object around in a
uniform electric field, the force on it remains
𝐹
constant. Because 𝐸 = 𝑄, this means that the value
of E is the same everywhere.
The strength of a uniform electric field is a measure of how rapidly the potential changes.
The equation which describes this divides the potential difference by the distance over
which the potential difference exists:
𝐸=
𝑉
𝑑
Where V is the potential difference between the oppositely charged parallel plates or
surfaces producing the electric field and d is the separation of the surfaces.
Equipotentials
As we move through an electric field, the electrical potential changes from place to place.
Those locations which all have the same potential can be connected by lines called
equipotential. The field will always be perpendicular to the equipotential lines, as a field is
defined as a region which changes the potential. Remember that field lines never cross
each other.
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4 RADIAL FIELDS
In the region around a positively charged sphere, or a point charge
like a proton, the electric field will act outwards in all directions away
from the centre of the sphere. The arrows in the diagram get further
apart as you move further away from the sphere, indicating that the
field strength reduces as you move away from the centre. This means
that the distance between equipotentials also increases. The field is
the means by which the potential changes, so if it is weaker, then the
potential changes less quickly.
5 COMBINATION ELECTRIC FIELDS
In a region where there are electric fields caused by more than one charged object, the
overall field is the vector sum at each point of the contributions from each field.
Charge is particularly concentrated in regions around spikes or points on charged objects.
The field lines are close together at these places and the field will be strong around them.
HSW – This is why lightning conductors are spiked. The concentrated charge at the point will
attract the lightning more strongly.
6 CHARGE PARTICLE INTERACTIONS – COULOMB’S LAW
The attraction between a proton and an electron can be imagined as the proton creating an
electric field because of its positive charge, and the electron feeling a force produced by the
proton’s field.
The basic law describing the size of the force F between two point charges Q1 and Q2, which
are separated by a distance r, is described by Coulomb’s law and is given by the inversesquare law relationship:
𝐹=
𝑘𝑄1 𝑄2
𝑟2
1
Where 𝑘 = 4𝜋𝜀 = 8.99 × 109 N m2 C-2. Here 𝜀0 is a constant known as the permittivity of
0
space, which is a measure of how easy it is for an electric field to pass through space.
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Radial field calculations
We have seen that an electric field can be defined as a region of space which will produce a
force on a charged particle. This definition allows us to come up with an expression for the
strength of a radial electric field using the expressions for force on a charged particle met
earlier:
𝐹𝑐𝑜𝑢𝑙𝑜𝑚𝑏 = 𝐸1 𝑄2 =
𝑘𝑄1 𝑄2
𝑟2
Thus, the radial field strength at a distance r from a charge Q is given by:
𝐸=
𝑘𝑄1
𝑟2
There are, of course, differences between
the two phenomena. One obvious one is
that gravity is about masses but electricity
is about charges. Other differences
include:
HSW – Powers of ten
Gravitational forces affect all
particles with mass, but
electrostatic forces affect only
particles that carry charge.
Gravitational forces are always
attractive but electrostatic forces
can be either attractive or
repulsive.
It is not possible to shield a mass
from a gravitational field but it is
possible to shield a charge from an
electrostatic field.
HSW – Comparing gravitational and
electric fields
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7 CAPACITANCE
Circuit symbol of a capacitor:
Electric fields in circuits
An electric field can cause charged particles to move. Indeed, this is why a current flows
through a circuit – an electric field is set up within the conducting material and this causes
electrons to feel a force and thus move through the wires and components of the circuit.
Where there is a gap in a circuit, although the effect of the electric field can be felt by
charges across empty space, conduction electrons are generally unable to escape their
conductor and move across a gap. This is why a complete path is needed for a simple electric
circuit to function.
However, charge can be made to flow in an incomplete circuit. When the power supply is
connected, the electric field created in the conducting wires, causes electrons flow towards
the positive terminal. Since the electrons cannot cross the gap between the plates they build
up on the plate connected to the negative terminal, which becomes negatively charged.
Electrons in the plate connected to the positive terminal flow towards the positive terminal,
resulting in a positive charge on that plate. The attraction between opposite charges across
the gap creates an electric field between the plates, which increases until the pd across the
plates is equal to the pd of the power supply.
A pair of plates like this with an insulator between them is called a capacitor. Charge will
build up on a capacitor until the pd across the plates equals that provided by the power
supply to which it is connected. At that stage it is said to be fully charged. The capacitor is
acting as a store of charge. The amount of charge a capacitor can store, per volt applied
across it, is called its capacitance, C, and is measured in farads (F). The capacitance depends
on the size of the plates, their separation, and the nature of the insulator between them.
Capacitance can be calculated from the equation:
𝐶=
𝑄
𝑉
Energy stored on a charged capacitor
A charged capacitor is a store of electrical potential energy. When the capacitor is
discharged, this energy can be transferred into other forms.
𝐸 = 𝑄𝑉
However, the energy stored in a charged capacitor is given by:
𝐸=
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𝑄𝑉
2
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In order to charge a capacitor, it begins with zero charge stored on it and slowly fills
up as the pd increases, until the charge at voltage V is given by Q. Each time extra
charge is added, it has to be done by increasing the voltage and pushing the charge
on, which requires energy. Work is done against the repulsion between like charges.
Q=CV, so we can use this to find two other versions of the equation for stored energy:
𝐸=
1
𝑄𝑉
2
1
= 𝐸 = 2 (𝐶𝑉)𝑉
1
=𝐸 = 2 𝐶𝑉 2
Or
1
𝑄
1 𝑄2
𝐸 = 𝑄( ) =
2
𝐶
2 𝐶
8 CAPACITOR DISCHARGE
If the two way-switch is moved to position B, the electrons on the
capacitor will be able to move under the influence of the electric
field towards the positive side of the capacitor. To do this they will
flow through the lamp.
At first, the rush of electrons as the capacitor discharges is as high as
it can be – the current starts at a maximum. As electrons flow from the discharging
capacitor, the pd across it is reduced and the electric field and hence the push on the
remaining electrons is weaker. The current is less. Eventually, the capacitor will be fully
discharged and there will be no more electrons moving from one side of the capacitor to the
other – the current will be zero. The discharging current, pd across the capacitor, and charge
remaining on the capacitor will follow the patterns as shown below:
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Increasing the time of discharge:
There are two possibilities. For the same maximum pd, increasing the capacitance, C, will
increase the charge stored, as Q = CV. Alternatively, the charge would flow more slowly if
the resistance, R, in the lamp circuit was greater.
An overall impression of the rate of discharge of a capacitor can be gained by working out
the time constant, 𝜏 = 𝑅𝐶, and with resistance in ohms and capacitance in farads, the
answer is in seconds. In fact, the time constant tells you how many seconds it takes for the
current to fall to 37% of its starting value.
By considering the charging process in the same way, it is possible to work out that the
charging process produces similar graphs.
When charging through a resistor, the time constant RC has exactly the same implications. A
greater resistance, or a larger capacitance, or both, means the circuit will take longer to
charge up the capacitor.
9 DISCHARGING CAPACITOR MATHS
The charging and discharging of a capacitor follows curved graphs in which the current is
constantly changing, and so the rate of change of charge and pd are also constantly
changing. These graphs are known as exponential curves. The shapes can be produced by
plotting mathematical formulae which have power functions in them. In the case of
discharging a capacitor, C, through a resistor, R, the function which describes the charge
remaining on the capacitor, Q, at a time, t, is:
𝑡
𝑄 = 𝑄0 𝑒 −𝑅𝐶
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The pd across a discharging capacitor will fall as the charge stored falls. By substituting the
equation Q = CV into our exponential decay equation, we can show that the formula that
describes voltage on a discharging capacitor has exactly the same form as that for the charge
itself:
𝑡
𝑄 = 𝑄0 𝑒 −𝑅𝐶 and Q = CV
𝑡
𝐶𝑉 = 𝐶𝑉0 𝑒 −𝑅𝐶
From which the capacitance term, C, can be cancelled, leaving:
𝑡
𝑉 = 𝑉0 𝑒 −𝑅𝐶
The discharging current also dies away following an exponential curve. Ohm’s law tells us
that V = IR, and hence 𝑉0 = 𝐼0 𝑅.
𝑡
𝐼𝑅 = 𝐼0 𝑅𝑒 −𝑅𝐶
From which the resistance term, R will cancel on both sides:
𝑡
𝐼 = 𝐼0 𝑒 −𝑅𝐶
10 MAGNETIC FIELDS
A magnetic field is a region in which magnetic materials or moving electrical charges
experience a force.
Magnetic lines are used just as electric lines to illustrate the direction and strength of
magnetic fields.
There is a simple convention for the N and S labels on the magnets and for the arrows on
the lines of the force.
Magnetic lines of force flow from N to S.
N and S are called the poles of the magnet.
Like electric fields, the magnetic field is strongest, that it exerts the largest forces, at
places where the magnetic field lines are most closely bunched.
Unlike positive and negative electric charges, an isolated N or S pole has never been
found.
Electric and magnetic fields are in fact very similar as seen and utterly intertwined, as
Maxwell’s work on the nature of electromagnetic radiation taught us.
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Electric currents also produce magnetic fields. Magnetic lines of force in these magnetic
fields are always closed loops. The field with the simplest shape is that produced by a steady
current in a long straight wire.
Current out of page
Current into page
The magnetic field lines are a series of concentric circles; the field is getting weaker further
from the wire.
Magnetic field of current-carrying coil
11 MAGNETIC FIELD STRENGTH
When a current-carrying wire is placed at right angles to a uniform magnetic field, the
magnetic fields interact, resulting in a force F on the wire. This force depends on the current
I in the wire and the length l of the wire that lies in the field:
𝐹 ∝ 𝐼𝑙
The strength of the field, which is called the magnetic flux density B, is a vector quantity, and
is defined as the constant of proportionality, so:
𝐹 = 𝐵⊥ 𝐼𝑙
The suffix ⊥ indicates that the wire carrying the current must be perpendicular to the
magnetic field. This equation can be written as:
𝐹 = 𝐵𝐼𝑙𝑠𝑖𝑛𝜃
Where 𝐵⊥ = 𝐵𝑠𝑖𝑛𝜃 and 𝜃 is the angle between B and the wire.
The unit N A-1 m-1 emerges from the calculation. This unit is usually given the name tesla,
symbol T.
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A consequence of the expression 𝐹 = 𝐵⊥ 𝐼𝑙 is that a motor can be made more powerful, or
faster by:
Increasing the current through the motor
Increasing the number of turns of wire in the motor
Increasing the magnetic field within the motor
12 FLEMING’S LEFT HAND RULE
Like gravitational field strength, and electric field strength, magnetic
field strength, B, is a vector quantity. But unlike gravitational and
electric fields, where the forces are parallel to the fields, the
magnetic force is perpendicular to both the field and the currentcarrying wire. Left-hand rule is used to illustrate this in 3D.
Remember that the left hand rule applies to the sense of
conventional current, or to the sense of motion of positive charge, and not to the direction
of motion of negatively charged electrons, which will be in the opposite direction.
13 F = BQVSIN𝜽
The current I in the wire is the result of the drift of very large numbers of electrons in the
wire. The relationship linking the current to the drift speed of the charged particles is:
𝐼 = 𝑛𝐴𝑣𝑞
Where n is the number of charge carriers per unit volume, A is the cross-sectional area of
the wire, q is the charge on each charge carrier and v is the drift speed of the charge carriers.
As the force on a wire is given by 𝐹 = 𝐵⊥ 𝐼𝑙, inserting 𝐼 = 𝑛𝐴𝑣𝑞 gives 𝐹 = 𝐵⊥ (𝑛𝐴𝑣𝑞)𝑙, which
can be rearranged as:
𝐹 = 𝐵⊥ 𝑣𝑞 × 𝑛𝐴𝑙
In this equation Al is the volume of the wire in the magnetic field, so nAl is the total number
N of charge carriers in that piece of wire.
𝐹
𝐹
Hence 𝑛𝐴𝑙 is equal to 𝑁, that is, it is the force on one of these charge carriers.
The result is that the force F on one charged particle moving at speed v perpendicular to a
magnetic field of flux density B is given by:
𝐹 = 𝐵⊥ 𝑞𝑣
Hence, 𝐹 = 𝐵𝑞𝑣𝑠𝑖𝑛𝜃
As the force on a charged particle is always at right angles to the direction of its velocity, it
acts as a centripetal force, and the particle follows a circular path. This means that, given the
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right combination of conditions, a moving charged particle could be held in place by a
magnetic field, continuously orbiting a central point. This is the principle by which artificially
generated antimatter is contained to save it from annihilation, for future use or study.
The force on a charged particle moving at right angles to a magnetic field is given by:
𝐹 = 𝐵⊥ 𝑞𝑣
Centripetal force is given by:
𝑚𝑣 2
𝐹=
𝑟
Combining these gives:
𝑟=
𝑚𝑣
𝑝
=
𝐵𝑞 𝐵𝑞
14 ELECTROMAGNETIC INDUCTION
The movement of a charged particle in a magnetic field causes it to feel a force. Newton’s
third law of motion states that this force must have a counterpart which acts equally in the
opposite direction. Moreover, this pair of electromagnetic forces is generated whenever
there is relative motion between a charge and a magnetic field. Thus, a magnetic field
moving past a stationary charge will create the same force. The velocity term in the
expression 𝐹 = 𝐵⊥ 𝑞𝑣 actually refers to the relative perpendicular velocity between the
magnetic field lines and q.
This means that if we move a magnet near a wire, the electrons in the wire will feel a force
tending to make them move through the wire. This is an emf – if the wire is in a complete
circuit, then the electrons will move, forming an electric current. This is one principle by
which we generate electricity. Reversing the direction of the magnetic field, or the direction
of the relative motion will reverse the direction of the force on the electrons, reversing the
polarity of emf.
15 FLUX LINKAGE
The product of magnetic flux density and the area through which it acts is called the
magnetic flux through the area, symbol Φ.
Φ = B⊥ A
However, for a coil with N number of turns this equation becomes:
Φ = NB⊥ A
As 𝐹 = 𝐵⊥ 𝑞𝑣, it is no surprise that the faster the relative motion between a magnetic field
and a conductor, the greater the induced emf. Faraday investigated this and determined a
law on matter. Faraday’s law of electromagnetic induction states that:
The magnitude of an induced emf is proportional to the rate of change of flux linkage.
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16 LENZ’S LAW
Lenz’s law states that the induced emf must cause a current to flow in such a direction as
to oppose the change in flux linkage that produces it, otherwise energy would appear from
nowhere.
17 CALCULATING INDUCED EMFS
Putting Faraday’s and Lenz’s laws together gives us an expression for calculating an induced emf:
𝜀=
−𝑑(𝑁Φ)
𝑑𝑡
Faraday’s law told us that the emf would be proportional to the rate of change of flux linkage. The
minus sign in the equation comes from Lenz’s law, to indicate the opposing direction.
HSW and RECALL from IGCSE
DC motors
As a current passes through a coil in a magnetic field, one side of it will experience
and upward force while the other side experiences a downward force. So there is a
turning moment in the coil. Commutator changes the direction of the current once in
a half turn, so rotation is continuous.
In practical motors:
The permanent magnets are replaced with electromagnets.
Single loop is replaced with several coils of wire wrapped on the same axes.
The coils are wrapped on a laminated iron core.
To increase the rate of turning of motors:
Increase the current
Increase the number of turns in coil
Increase magnetic field
Electromagnetic induction
When a wire is moved in a magnetic field, a voltage is induced. If the wire is a part of
complete circuit, an induced current passes through it. This is called electromagnetic
induction.
To increase the size of induced voltage:
Move wires faster
Increase the number of turns in coil
Increase magnetic field
Cross-sectional area of the coil can be increased.
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Faraday’s law of electromagnetic induction
The size of the induced voltage across the ends of wire is directly proportional to the rate at
which the magnetic lines of flux are being cut.
Lenz’s law
When a current is induced, it always opposes the change in magnetic field.
To increase the size of induced current:
Move magnet faster
Increase the number of turns in the solenoid
Use stronger magnet
Use a thicker iron core
Loudspeakers
Motor effect is applied to loudspeakers.
A coil is attached to a paper cone. The changing current in the coil produces a
changing magnetic field which interacts with the field from the permanent magnet.
This creates backwards and forwards motion of the coil and paper cone.
This makes the air vibrate and sound waves are generated.
Electric generator
As the coil rotates, its wires cut through magnetic field lines and a current is induced
in them. The current induced in the coil flows first in one direction and then in
opposite direction. This kind of current is alternating current. It changes direction
once every half turn.
To increase the size of pd:
Turning the coil faster
Using stronger magnets
Increasing the number of turns
Using an iron core
Transformers
A transformer changes the size of an alternating voltage by having different number of turns
on the primary coil and secondary coil. Alternating current in the primary coil produces an
alternating magnetic field. Iron core is magnetised and links the alternating magnetic field to
the secondary coil. An alternating current is induced in the secondary coil.
𝑉𝑝 𝑛𝑝
=
𝑉𝑠 𝑛𝑠
Power input equals the power output in ideal transformers. Therefore,
𝑉𝑝 𝐼𝑝 = 𝑉𝑠 𝐼𝑠
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Particle physics
1 ALPHA PARTICLE SCATTERING
Geiger and Marsden undertook an experiment in which they
aimed an alpha particle source at an extremely thin gold foil.
Their expectation was that all the alpha particles would pass
through, possibly with little deviation. The results generally
followed this pattern – the vast majority passed straight through
the foil. However, a few alpha particles had their trajectories
deviated by quite large angles. Some were even repelled back the
way they had come.
2 NUCLEAR STRUCTURE
The nucleus made up from two particles: the proton and the neutron. Collectively these two
particles, when in a nucleus, are known as nucleons. The number of protons in a nucleus
determines which element the atom will be. The periodic table is a list of the elements
ordered according to the number of protons in each atom’s nucleus. This number is called
the proton number or the atomic number. For small nuclei, the number of neutrons in the
nucleus is generally equal to the number of protons. Above atomic number 20, for the
nucleus to be stable more neutrons than protons are generally needed. The neutrons help to
bind the nucleus together as they exert a strong nuclear force on other nucleons, and they
act as a space buffer between the mutually repelling positive charges of the protons.
3 ELECTRON BEAMS
Free conduction electrons in metals need a certain amount of energy if they are to escape
from the surface of the metal structure. The electrons can gain enough energy through
heating of the metal. The release of electrons from the surface of a metal as it is heated is
known as thermionic emission.
If, on escaping, these electrons find themselves in an electric field, they will be accelerated
by the field, moving in the positive direction. The kinetic energy they gain will depend on the
pd, V, that they move through, according to the equation:
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𝐸𝑘 = 𝑒𝑉
Using thermionic emission to produce electrons, and applying an electric field to accelerate
them, we can generate a beam of fast-moving electrons, traditionally known as cathode ray.
This beam of electrons will be deflected by a magnetic field. If a fast-moving electron hits a
screen painted with a certain chemical, the screen will fluoresce – it will emit light.
4 ELECTRONS AS WAVES
Electrons do not just behave as particles – they also have wave properties. De Broglie’s wave
equation relates the wavelength of a particle to its momentum:
𝜆=
ℎ
𝑝
The idea of electrons acting as waves has allowed scientists to study the structure of crystals.
When waves pass through a gap which is about the same size as their wavelength, they are
diffracted – they spread out. The degree of diffraction spreading depends on the ratio of the
size of the gap to the wavelength of the wave. Electron diffraction and alpha particle
scattering both highlight the idea that we can study the structure of matter by probing it
with beams of high-energy particles. The more detail – or smaller scale – the structure to be
investigated, the higher energy the beam of particles needs to be.
5 PARTICLE ACCELERATORS
To investigate the internal substructure of particles, scientists collide them with other
particles at very high speeds (very high energies). If we can collide particles together hard
enough they will break up, revealing their structure. In most cases additional particles are
created from the energy of the collision.
The challenge for scientists has been to accelerate particles to sufficient speeds. Charged
particles can be accelerated in straight lines using a potential difference, and their direction
changed along a curved path by a magnetic field.
Linear accelerators
A linear accelerator operates on the same principle as an electron gun – electrons or other
charged particles are accelerated across gaps between charged electrodes. In a linac, there
often many drift tubes connected to a high-frequency, high-voltage AC supply. These are
arranged in such a way that the particles gain kinetic energy between the tubes and move at
constant speed inside the tubes. The main principles are:
Alternate tubes are connected to each terminal of the AC supply.
The charged particles spend one half of each period of the alternating voltage
between two tubes and the other half of each cycle inside one of the tubes.
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During a half-cycle when the voltage would oppose their motion, the particles are
inside a tube, where they are shielded from the electric field; the particles therefore
travel at constant speed within the tube.
The particles gain kinetic energy as they travel across successive gaps, and can be
accelerated to high energies.
The length of the drift tubes increases along the accelerator so that although the
speed of the particles is increasing, the time needed to pass through each tube will
always be the same (equal to the half period of the alternating electric field).
Cyclotron
The main disadvantage of linear accelerators is that, in order to
produce high energies, they need to be very long. Cyclotrons,
while based on the same principle of synchronous acceleration
as linacs, use a magnetic field to make the charged particles
move in a spiral path.
Accelerating particles in circles
By Fleming’s left-hand rule, a particle carrying a charge q
moving with speed v at right angles to a magnetic field of flux
density B will experience a force of magnitude Bqv in a
direction perpendicular to its motion. The particle will therefore follow a circular path, and
we have:
The main principles of a cyclotron are:
A cyclotron consists of two hollow, semicircular D-shaped sections which are placed at right
angles to a uniform magnetic field and have a high-frequency alternating voltage applied
between them.
An ion source fires charged particles into the gap between the dees close to the
centre.
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The particles are accelerated across the gap during a half-cycle of the alternating
voltage when the polarity is appropriate.
During the next half-cycle, the particles follow a circular path at constant speed in
one of the dees.
The particles are then accelerated across the gap into the other dee.
Faster particles follow paths of greater radius, so that all particles spend the same
amount of time in the dees and their acceleration is synchronised with the
appropriate half-cycle of alternating voltage.
Relativistic effects
For speeds approaching the speed of light, relativistic effects need to be taken into account.
It is a basic postulate of the theory of relativity that nothing can travel beyond the speed of
light. Therefore, for energy to be conserved, the mass of the electron must increase.
Relativistic effects can therefore create synchronisation problems in high-energy particle
accelerators. Synchrotrons, such as CERN, account for these relativistic effects and can
produce particles with extremely high energy.
6 PARTICLE DETECTORS
Particles can be detected when they interact with matter to cause ionisation or when they
excite electrons to higher energy levels, accompanied by the emission of photons.
In bubble tanks and cloud chambers, a charged particle passing through will generate a trail
of ions along which bubbles or vapour droplets are formed, making these paths visible. The
nature of the particles can be deduced from the length of trails they leave, and from how
these paths are affected by electric and magnetic fields.
Interpreting images
Neutral particles do not produce a trail and cannot be observed.
Oppositely charged particles are deviated in opposite directions, which is
determined by Fleming’s left-hand rule.
From the relation r = p / Bq, we can infer that particles with greater momentum
spiral less, so trails with greater radius indicate massive and/or energetic particles.
A thicker trail indicates more intense ionisation, which is usually due to the particle
having greater momentum or charge.
Particles spiral inwards. This is because, as they lose energy, their momentum
decreases and the radius of the path become smaller.
7 PARTICLE INTERACTIONS
In any interaction:
Momentum is conserved.
Mass/energy is conserved.
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Charge is conserved.
Baryon number is conserved.
Lepton number is conserved.
Strangeness is conserved.
Conservation of mass-energy
The rest mass of particles can be represented as an equivalent energy by means of Einstein’s
equation:
𝐸
𝐸0 = 𝑚0 𝑐 2 or 𝑚𝑜 = 𝑐 02
The rest mass can be thought of as the energy that would be transferred if the entire mass
were to be dematerialised, or if the particle were to be made up completely from other
forms of energy.
Remember mass of particles moving close the speed of light are greater than their rest mass.
This is because nothing can travel beyond the speed of light, thus for momentum to be
conserved mass must increase.
The mass of moving particles can be taken as being the same as their rest mass unless the
sped is close to the speed of light.
Units of mass and energy
It is often convenient to represent the rest mass of subatomic particles in terms of the nonSI units MeV/c2 or GeV/c2.
MeV and GeV are units of energy; MeV/c2 and GeV/c2 are the corresponding units of mass.
Another unit used in particle physics is the unified atomic mass unit, denoted by u. It is
defined as one-twelfth of the mass of a carbon-12 atom, so
1u = 1.66 x 10-27kg
8 CREATION AND ANNIHILATION OF MATTER AND ANTIMATTER
Early work with particle detectors showed that cosmic rays could produce some tracks
identical to those of an electron, but which curve in the opposite direction. This was the first
piece of evidence for the existence of antimatter. Energy of photons was creating particles in
a process known as pair production. This is called creation. Conversely, if a particle and
antiparticle meet, they will spontaneously vanish from existence to be replaced by the
equivalent energy. This is called annihilation. Einstein’s famous equation is used to calculate
the energy or mass created as a result of these processes.
∆𝐸 = 𝑐 2 ∆𝑚
In order to create a new particle the energy converted must be at least equal to the rest
mass energy of the particle. If there is more energy converted, the new particle will gain
some kinetic energy.
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9 THE STANDARD MODEL
Our current ideas about fundamental particles are summarised by the standard model of
particle theory. In this model there are two types of fundamental particle – quarks and
leptons.
1-) Quarks are strongly interacting particles, and it is believed they do not exist singly. They
occur in two possible combinations.
In quark – anti-quark pairs, called mesons
In quark triplets, called baryons.
Collectively mesons and baryons are referred to as hadrons. The proton and the neutron are
familiar examples of baryons.
2-) Leptons are weakly interacting particles. Leptons occur singly; the electron to a familiar
example of a lepton.
In the standard model there are six quarks and six leptons, plus their antiparticle
equivalents.
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10 WAVE-PARTICLE DUALITY - AGAIN
Diffraction patterns can be observed when high-energy electrons from an electron gun are
fired through a thin slice of carbon, indicating that the electron, as well as being a particle,
behaves like a wave. It is also possible to measure the radiation pressure of light from the
Sun, which shows that photons have momentum, an attribute to all particles. This waveparticle duality applies to all particles, but it is significant only on a subatomic scale. The
relationship between momentum p of a particle and its wavelength 𝜆 is expressed by de
Broglie’s wave equation:
𝜆=
ℎ
𝑝
where h is the Planck constant.
In the electron diffraction experiment it was also observed that when the voltage across the
electron gun is increased, the higher-energy electrons produced have a shorter wavelength.
This relationship can be seen from Planck’s photon equation.
𝐸=
ℎ𝑐
𝜆
The wavelength of a 100 keV electron is of the same order of magnitude as the diameter of a
carbon atom, so a noticeable diffraction pattern is produced when an electron beam is
focused on a specimen containing carbon. This principle is applied to electron microscopes.
If a high-energy accelerator is used, the wavelength of the electrons will be much shorter,
and this enables the structures of hadrons to be probed using deep inelastic scattering.
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